c、printf、 double : getting more than 15 decimal digits

Question

我正在实现一个简单的算法来估计 c 中 double 的 sqrt，类似于我在 java 中学习的方法。

但是，当我使用 printf("当前猜测:%.30f\n",guess); 时为了打印我当前的猜测，我实际上得到了 30 位 double 。我的印象是，由于 IEEE， double 只能有 15 位十进制数字？这些数字从哪里来？

我的完整代码:

int main() {
// Finding the sqrt of x
double x = 1337.;
double high = x;
double low = 0.;
double guess;
double lastguess = -1.;
for(int i=0; i<1000; i++) {
    lastguess = guess;
    guess = low+(high-low)/2;
    if(compDouble(guess, lastguess)) {
        break;
    }
    if(guess*guess > x) {
        // Guess was too high
        // get the lower interval
        low = low;
        high = guess;
    } else {
        // Guess was too low, get the upper interval
        low = guess;
        high = high;
    }
    printf("Current guess: %.30f \n", guess);
}
printf("sqrt of %f is about %f", x, guess);
return 1;
}

输出:

Current guess: 668.500000000000000000000000000000 
Current guess: 334.250000000000000000000000000000 
Current guess: 167.125000000000000000000000000000 
Current guess: 83.562500000000000000000000000000 
Current guess: 41.781250000000000000000000000000 
Current guess: 20.890625000000000000000000000000 
Current guess: 31.335937500000000000000000000000 
Current guess: 36.558593750000000000000000000000 
Current guess: 39.169921875000000000000000000000 
Current guess: 37.864257812500000000000000000000 
Current guess: 37.211425781250000000000000000000 
Current guess: 36.885009765625000000000000000000 
Current guess: 36.721801757812500000000000000000 
Current guess: 36.640197753906250000000000000000 
Current guess: 36.599395751953125000000000000000 
Current guess: 36.578994750976562500000000000000 
Current guess: 36.568794250488281250000000000000 
Current guess: 36.563694000244140625000000000000 
Current guess: 36.566244125366210937500000000000 
Current guess: 36.564969062805175781250000000000 
Current guess: 36.565606594085693359375000000000 
Current guess: 36.565287828445434570312500000000 
Current guess: 36.565128445625305175781250000000 
Current guess: 36.565048754215240478515625000000 
Current guess: 36.565008908510208129882812500000 
Current guess: 36.565028831362724304199218750000 
Current guess: 36.565018869936466217041015625000 
Current guess: 36.565013889223337173461914062500 
Current guess: 36.565011398866772651672363281250 
Current guess: 36.565010153688490390777587890625 
Current guess: 36.565010776277631521224975585938 
Current guess: 36.565010464983060956001281738281 
Current guess: 36.565010620630346238613128662109 
Current guess: 36.565010542806703597307205200195 
Current guess: 36.565010581718524917960166931152 
Current guess: 36.565010601174435578286647796631 
Current guess: 36.565010591446480248123407363892 
Current guess: 36.565010596310457913205027580261 
Current guess: 36.565010598742446745745837688446 
Current guess: 36.565010597526452329475432634354 
Current guess: 36.565010598134449537610635161400 
Current guess: 36.565010597830450933543033897877 
Current guess: 36.565010597678451631509233266115 
Current guess: 36.565010597602451980492332950234 
Current guess: 36.565010597564452154983882792294 
sqrt of 1337.000000 is about 36.565011

Answer 1

I was under the impression that due to IEEE, a double can only have 15 decimal digits?

嗯，这是错误的。

试试这个:

double d = 1.0e-45;
printf("d is %.60f \n", d);

输出:

d is 0.000000000000000000000000000000000000000000001000000000000000 

如您所见，小数位数远多于 15 位。

小数位数取决于数字的值。非常小的数字可以有很多小数位(前面有很多零)，而非常大的数字根本不能有任何小数位。有些非常大的数字甚至不能表示奇数。

标准 64 位 double 可以具有的最小非零(正)值为 2^-1074，即:

0.000......04940656.............47265625
            ^                          ^
            |                          |
            Decimal digit #324         Decimal digit #1074

使用以下代码尝试一下:

int main(void)
{
  assert(sizeof(double) == sizeof(uint64_t));

  union
  {
    uint64_t i;
    double d;
  } x;

  x.i = 1;  // The smallest non-zero positive double value
  printf("%.1100f \n", x.d);

  return 0;
}